The local (in time) well-posedness of (REF )
in \(H^3\)  can be shown via standard procedures (see, e.g., [1]}). It suffices to
establish the global bounds stated in Theorem REF  in order to obtain the
global existence. This is accomplished by applying the bootstrapping argument to (REF ),
namely
\(E(t)\le C_1F(u_0,b_0)+ C_2\Big (E^{\frac{3}{2}}(t)+E^2(t)\Big ),\) 

where
\(F(u_0,b_0)&=\Vert (u_0,b_0)\Vert _{H^3}^2+\Vert (u_0,b_0)\Vert _{L^2_{x_3}L^1_{x_1x_2}}^2+\Vert (\partial _3u_0,\partial _3b_0)\Vert _{L^2_{x_3}L^1_{x_1x_2}}^2\\&\quad +\Vert (\partial _3^2u_0,\partial _3^2b_0)\Vert _{L^2_{x_3}L^1_{x_1x_2}}^2.\) 

A useful description of the bootstrapping argument can be found in [2]}.
In order to apply the bootstrapping argument, we make the ansatz that
\(E(t)\le M:=\min \Big \lbrace 1,\frac{1}{(4C_2)^2}\Big \rbrace . \) 

We then verify that \(E(t)\)  actually admits a smaller bound,
\(E(t)\le \frac{M}{2}.\) 

Inserting (REF ) in (REF ) and recalling the initial assumption (REF ), we have
\(E(t)&\le C_1F(u_0,b_0)+C_2\Big ( M^\frac{1}{2} + M \Big )E(t)\\&\le C_1 \delta ^2 + 2C_2M^{\frac{1}{2}}E(t)\\&\le C_1 \delta ^2 + \frac{1}{2} E(t),\) 

or
\(E(t)\le 2C_1 \delta ^2.\) 

If the initial data is sufficiently small, say
\(\delta ^2 \le \frac{M}{4C_1},\) 
then we derive
\(E(t)\le 2C_1\delta ^2 \le \frac{M}{2}.\) 

The bootstrapping argument then implies \(T=\infty \)  and asserts that for any time \(t>0\) ,
\(E(t)\le C\delta ^2,\) 

which, in particular, implies the desired global bound on the solution \((u,b)\) . As a
consequence, we obtain the global existence of solutions. The uniqueness is obvious due to the
high regularity of the solution. The global bound on \(E_2(t)\)  yields the desired decay rates
stated in Theorem REF . This completes the proof of Theorem REF .
